Group Theory

Group theory is the notion of abstraction that are common to a number of key areas being studied essentially simultaneously.The three main areas that pose the theory of groups are:(1) the geometry of the early 19th century,(2) the theory of numbers in the late 18th century,(3) the theory of algebraic equations in the late 18th century that led to the study of permutations.
(1) The geometry has been studied for a very long time so it is natural to ask what happens to the geometry of the early 19th century who contributed to the increase in the group concept.Geometry has begun to lose the 'metric' his characters with projective geometry and non-euclidean being studied. Also a movement to learn geometry in n dimensions leads to abstraction in geometry itself. The difference between the incidence geometry and metrics derived from the work of Monge, Carnot students and perhaps the most important work of Poncelet. Non-euclidean geometry studied by Lambert, Gauss, Lobachevsky and János Bolyai, among others.Möbius in 1827, although he is really aware of the concept of the group, began to classify the geometry use the fact that the particular geometry studies the invariant properties under a certain group. Steiner in 1832 studying synthetic understanding of geometry which eventually became part of the research group of transformations.
(2) In 1761 Euler studied modular arithmetic. In particular, he examined the rest of the power of n modulo number. Although Euler's work ', of course, is not expressed in theoretical terms the group he did not give examples of abelian groups into cohimpunans decomposition of a subgroup. He also proved a special case of a sequence of subgroups of a divisor of the order of the group.Gauss in 1801 is to take the Euler job 'even further and give quite a lot of work on modular arithmetic which amounts to pretty much the theory of abelian groups. She checked the command element and prove (although not in this notation) that there is a sub for himpunaniap cyclic group of order dividing numbers. Gauss also examined other abelian groups. He looked at the binary quadratic formax 2 + 2 bxy + cy 2 where a, b, c are integers.Gauss examine the behavior of the transformation and substitution. He forms a partition into classes and then determine the composition of the class. Gaussmembuktikan that the sequence composition of the three forms is a material that, in modern language, the associative law applies. In fact, Gauss had a finite abelian group and then (in 1869).
(3) permutation was first studied by Lagrange in his paper in 1770 on the theory of algebraic equations. Lagrange's main object is to find out why the cubic and quartic equations can be solved algebraically. In studying the cubic, for example, Lagrange assumes the cubic root of the given equation is x ', x''and x'''.Then, take 1, w, w ^ 2 as the cube root of unity, he examined the expressionR = x '+ w + w ^ 2''x'''and note that it takes only two different values ​​under the six permutations of the roots x ', x'', x'''. Although the initial permutation group theory can be seen in this work, Lagrange never composes his permutations, so in some ways the group never discussed at all.
The first person who claims that the equation of degree 5 can not be solved algebraically is Ruffini. In 1799 he published works whose purpose is to show insolubility general quintic equation.Ruffini's work 'is based on that of Lagrange but Ruffini introduce permutation group. This he called permutation and explicitly using the closure properties (associative law is always applicable to permutation). Ruffini permutazione divide into types, namely semplice permutation group which is cyclic in modern notation, and composta permutation groups of non-cyclic.Permutation composta The Ruffini divided into three types in the current notation is intransitive group, transitive imprimitive groups and transitive primitive groups.
Ruffini evidence of this disappointing with the lack of reaction to it, the paper Ruffini published further evidence. In a 1802 paper he shows that the permutation group associated with a reduced equation takes transitive understanding well beyond that of Lagrange.
Cauchy played a major role in developing the theory of permutations. His first paper on the subject was in 1815 but at that stage iniCauchy motivated by a permutation of the roots of the equation. However, in 1844, Cauchy published a major work that make up the theory of permutations as a subject in its own right. He introduced the notation of power, positive and negative, permutation (with power 0 gives the identity permutation), defines the order of a permutation, cycle and use the notation introduced the term Systeme des conjuguées substitution group. Cauchy calls two permutations of the same if they have the same cycle structure and prove that this is the same as the permutation which conjugates.
Abel, in 1824, provides the first evidence received from the insolubility of the quintic, and he uses the ideas that exist in the permutation of the roots but little new in the development of group theory.
Galois in 1831 was the first to really understand that the solution of an algebraic equation is related to the structure le Groupe permutation group related to the equation. By 1832 Galois had found that a special subgroup (now called a normal subgroup) is fundamental. He called the group decomposition into sub cohimpunans decomposition of the right if the right and left simultaneously cohimpunan decomposition. Galois then showed that the non-abelian simple groups having the smallest order is the order of 60.
Galois job is unknown until Liouville published Galois paper in 1846. Liouville see clearly the connection between the theory of permutations Cauchy and Galois job. Liouville but failed to understand that the importance of work lies in the concept of Galois groups.
Betti started in 1851 published his theory of permutations and the associated theory of equations. Even Betti was the first to prove that the Galois group associated with the equation is actually a group of permutations in the modern sense. Serret published an important work discusses Galois' work, still without looking at the importance of the group.
Jordan in 1869 and 1870 papers from 1865 show that he was aware of the importance of the permutation group. He defines the permutation group isomorphism and prove Jordan - Holder theorem for permutation groups. Holder was to prove in the context of an abstract group in 1889.
Klein's Erlangen program proposed in 1872 which is a geometric group theory classification. The group would become center stage in mathematics.Perhaps the most remarkable developments came even before the Betti. This is due to the English mathematician Cayley. In early 1849 Cayley published a paper linking his ideas on permutations Cauchy. In 1854 Cayley wrote two papers are remarkable for their insight have an abstract group. At that time the only known group of permutation groups and even this is a radically new areas, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the Cayley table of some special permutation groups but, far more significant for the introduction of the concept of an abstract group, he realized that matrices and quaternions are a group.
Cayley paper about 1854 so far ahead of their time that they have little impact. But when Cayley back to the topic in 1878 with four papers on the group, one of them called group theory, the time is right for the concept of abstract group moves toward the center of mathematical inquiry. Cayley proved, among other results, that himpunaniap groups to be represented as a permutation group.Cayley's work prompted Holder, in 1893, to investigate groups of order p 3, pq 2, PQR and p 4.Frobenius and the Net (Kronecker students) bring forward the theory of groups. As far as the abstract concept is concerned, the next major contributor is the Von Dyck. Von Dyck, who had obtained his doctorate under Klein then became assistant to Klein's supervision. Von Dyck, with a fundamental paper in 1882 and 1883, was built for free groups and the definition of abstract groups in terms of generators and relations.
Group theory really come of age with a book by Burnside Theory of groups of order until published in 1897. The second volume of algebra book by Heinrich Weber (a student of Dedekind) Lehrbuch der Algebra published in 1895 and 1896 became a standard text. These books are influencing the next generation of mathematicians bring to the group theory may be a major part of the mathematical theory of the 20th century.
Sources:http://www-history.mcs.st-and.ac.uk/HistTopics/Development_group_theory.html